<h2>题目编号 : 228</h2>
<div style="color:#666;font-size:80%;">17 January 2009</div><br />
<div class="problem_content">
<p>Let <var>S</var><img src="" style="display:none;" alt="_(" /><sub>n</sub><img src="" style="display:none;" alt=")" /> be the regular <var>n</var>-sided polygon &ndash; or <i>shape</i> &ndash; whose vertices 

<var>v</var><img src="" style="display:none;" alt="_(" /><sub><var>k</var></sub><img src="" style="display:none;" alt=")" /> (<var>k</var>&thinsp;=&thinsp;1,2,&hellip;,<var>n</var>) have coordinates:</p>
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    <td><var>x</var><img src="" style="display:none;" alt="_(" /><sub><var>k</var></sub><img src="" style="display:none;" alt=")" />&nbsp;&thinsp;&nbsp;= &thinsp;
        cos( <img src="" style="display:none;" alt="^(" /><sup>2<var>k</var>-1</sup><img src="" style="display:none;" alt=")" />/<img src="" style="display:none;" alt="_(" /><sub><var>n</var></sub><img src="" style="display:none;" alt=")" /> <img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />180&deg;&thinsp;)</td>
  </tr><tr>
    <td width="40"></td>
    <td><var>y</var><img src="" style="display:none;" alt="_(" /><sub><var>k</var></sub><img src="" style="display:none;" alt=")" />&nbsp;&thinsp;&nbsp;= &nbsp;
        sin( <img src="" style="display:none;" alt="^(" /><sup>2<var>k</var>-1</sup><img src="" style="display:none;" alt=")" />/<img src="" style="display:none;" alt="_(" /><sub><var>n</var></sub><img src="" style="display:none;" alt=")" /> <img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />180&deg;&thinsp;)</td>
  </tr>
</table>

<p>Each <var>S</var><img src="" style="display:none;" alt="_(" /><sub><var>n</var></sub><img src="" style="display:none;" alt=")" /> is to be interpreted as a filled shape consisting of all points on the perimeter and in the interior.</p>

<p>The <i>Minkowski sum</i>, <var>S</var>+<var>T</var>, of two shapes <var>S</var> and <var>T</var> is the result of 

adding every point in <var>S</var> to every point in <var>T</var>, where point addition is performed coordinate-wise: 

(<var>u</var>,&thinsp;<var>v</var>) + (<var>x</var>,&thinsp;<var>y</var>) = (<var>u</var>+<var>x</var>,&thinsp;<var>v</var>+<var>y</var>).</p>

<p>For example, the sum of <var>S</var><img src="" style="display:none;" alt="_(" /><sub>3</sub><img src="" style="display:none;" alt=")" /> and <var>S</var><img src="" style="display:none;" alt="_(" /><sub>4</sub><img src="" style="display:none;" alt=")" /> is the six-sided shape shown in pink below:</p>

<div style="text-align:center;">
<img src="project/images/p_228.png" alt="picture showing S_3 + S_4" /></div>

<p>How many sides does <var>S</var><img src="" style="display:none;" alt="_(" /><sub>1864</sub><img src="" style="display:none;" alt=")" />&thinsp;+&thinsp;<var>S</var><img src="" style="display:none;" alt="_(" /><sub>1865</sub><img src="" style="display:none;" alt=")" />&thinsp;+&thinsp;&hellip;&thinsp;+&thinsp;<var>S</var><img src="" style="display:none;" alt="_(" /><sub>1909</sub><img src="" style="display:none;" alt=")" /> have?</p>
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